First note that the map \(e_* : H_n(H;\mathbb {Z}) \rightarrow H_n(H;\mathbb {Z})\)  induced by such an embedding is an isomorphism, as it respects the intersection form and the intersection form of \(H\)  is non-degenerate.
Given such an embedding, by pushing inwards along a collar we can suppose that it has image in the interior of \(H\) . Then the difference \(K := H \setminus \mathrm {int}(e(H))\)  is a cobordism from \(e(\partial H)\)  to \(H\) . It is easy to see using the Seifert–van Kampen theorem that \(K\)  is simply-connected, and by excision we have \(H_*(K, e(\partial H);\mathbb {Z}) \cong H_*(H, e(H);\mathbb {Z})\)  which vanishes by the discussion above. Thus \(K\)  is a simply-connected \(h\) -cobordism, so by the \(h\) -cobordism theorem [1]} there is a diffeomorphism \(K \cong [0,1] \times e(\partial H)\)  relative to \(e(\partial H)\) . Using this product structure, and a collar of \(H\) , we can find an isotopy from \(e\)  to a diffeomorphism.
